In the scene where he is tasked to deliver the message he gets into his Rover SD1 car. As his handler exits the car, he bangs the steering wheel center boss in anger and the car horn is sounded. This is an error as the horn switch on this vehicle was on the end of a stalk, not the steering wheel.
(at around 1h 40 mins) You can clearly see an poster ad for iPhone when they are driving. iPhones were not around at the time the movie is set.
When Marty is driving the taxi at night in the early part of the film a car can clearly be seen in front of his. This is supposed to be set in 1988 yet the car in front is a Ford which dates 2006 at least.
When Marty walks into the bakery to see his girlfriend and child, a 'Scores on the Doors' sign is visible on the door as he opens it - 'Scores on the Doors' didn't come into effect until 2005 while the events in the film are set in and around 1990.
After the kneecapping at the beginning of the film, the IRA drive off in a London cab with British number plates. Northern Irish ones have a different format.